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Full text of "Analytical Mechanics"

ivj.wJL.Lw.Ln
Substituting this value of G in the torque equation we have T doo        0 _,..      0
I-rr = - 2FDCOS-
dt                     2
—   h
FIRST APPROXIMATION. — When 0 and <£ are small the following relations give close enough approximations.
m      1                      0
T = img,    cos- = 1, 2
D
JDO ==: 6</>j      sin <j!> == 0== ~r 0.
Making these substitutions in the torque equation we get
du _ __ m^D2
which is the equation of simple harmonic motion.   Therefore
mg
is the period of the motion.
SECOND APPROXIMATION. — From Fig. 143 we have
FIG. 143.
and   ea'= Zsin</>= 2Dsin- -
-2
Therefore and
2D .   0 sin 0 = — sin ~
I              2i
T =
•sin2 ^
* The line ea' is considered as an arc of each of two circles with centers at g and c.